Theorem sbcieg 3769

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2152, ax-12 2189. (Revised by GG, 12-Oct-2024.)

Hypothesis

Ref	Expression
sbcieg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcieg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		df-sbc 3731	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		sbcieg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	elabg 3621	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	bitrid 284	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  {cab 2718  [wsbc 3730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-sbc 3731

This theorem is referenced by:  sbcie  3771  2nreu  4379  reuprg0  4641  rabsnif  4662  ralrnmptw  7042  ralrnmpt  7044  fpwwe2lem3  10554  nn1suc  12194  opfi1uzind  14471  mndind  18794  fgcl  23868  cfinfil  23883  csdfil  23884  supfil  23885  fin1aufil  23922  ifeqeqx  32637  nn0min  32920  bnj1452  35241  cdlemk35s  41436  cdlemk39s  41438  cdlemk42  41440  2nn0ind  43397  zindbi  43398  prproropreud  47991